Abstract
For 1<p<∞ and M the centered Hardy–Littlewood maximal operator on R, we consider whether there is some ε=ε(p)>0 such that ||Mf||p≥(1+ε)||f||p. We prove this for 1<p<2. For 2≤p<∞, we prove the inequality for indicator functions and for unimodal functions.
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